This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# SumConvergence

 SumConvergence gives conditions for the sum to be convergent. SumConvergencegives conditions for the multiple sum to be convergent.
• The following options can be given:
 Assumptions \$Assumptions assumptions to make about parameters Direction 1 direction of summation Method Automatic method to use for convergence testing
• Possible values for Method include:
 "IntegralTest" the integral test "RaabeTest" Raabe's test "RatioTest" D'Alembert ratio test "RootTest" Cauchy root test
• With the default setting Method, a number of additional tests specific to different classes of sequences are used.
• For multiple sums, convergence tests are performed for each independent variable.
Test for convergence of the sum :
Test the convergence of :
Find the condition for convergence of :
Test for convergence of the sum :
 Out[1]=
Test the convergence of :
 Out[2]=

Find the condition for convergence of :
 Out[1]=
 Scope   (14)
Exponential or geometric sums:
Plot the partial sums:
Polynomial exponential sums:
Rational sums:
Convergence picture:
Special functions:
Piecewise functions:
Slowly converging sums in the Abel-Dini scale:
Alternating sums:
Complex-valued sums:
Exponential or geometric series:
Parameter region for convergence:
Power series:
The convergence region for :
Combined series:
Piecewise sums:
Assuming to be complex:
A multivariate sum:
 Options   (4)
The ratio test typically applies to exponential and hypergeometric terms:
In this case the ratio test is inconclusive:
The root test typically applies to exponential terms:
In this case the root test is inconclusive:
The Raabe test works well for rational functions:
In this case the Raabe test is inconclusive:
The integral test works well on logarithmic terms:
In this case the integral test is inconclusive:
 Applications   (1)
Find the radius of convergence of a power series:
Prove convergence of Ramanujan's formula for :
Sum it:
Convergence properties are not affected by multiplication of constants:
Convergence is not affected by translating arguments:
SumConvergence is automatically called by Sum:
Many conditions generated by Sum are in effect convergence conditions:
With the setting VerifyConvergence->False, typically a regularized value is returned:
SumConvergence is used in sum transforms such as ZTransform:
Conditionally convergent periodic sums: